×

zbMATH — the first resource for mathematics

Real partitions of measure spaces. (English. Russian original) Zbl 0832.28012
Sib. Math. J. 35, No. 1, 189-191 (1994); translation from Sib. Mat. Zh. 35, No. 1, 207-209 (1994).
In this short note the author defines the notion of real partition and with its help existence is proved for measure-preserving mappings of several probability spaces onto the unit interval with the Lebesgue measure.
A real partition of a probability space \((X, {\mathcal A}, \mu)\) is defined to be a partition \((D_t)_{t \in [0,1]}\) of the space, such that \(\mu (A^-_t) = \mu (A^+_t) = t\) for all \(t \in [0,1]\), where \(A^-_t = \cup_{s < t} D_s\) and \(A^+_t = \cup_{s \leq t} D_s\).
The main theorem shows that: For every complete nonatomic probability space which includes a Cantor set, there exists a measure-preserving epimorphism of the space onto the unit interval with the Lebesgue measure.
MSC:
28A99 Classical measure theory
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. N. Berestovskii, ”Homogeneous manifolds with intrinsic metric. I,” Sibirsk. Mat. Zh.,29, No. 6, 17–29 (1988). · Zbl 0671.53036
[2] V. N. Berestovskii, ”Homogeneous spaces with intrinsic metric,” Dokl. Akad. Nauk SSSR,301, No. 2, 268–271 (1988).
[3] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).
[4] H. Busemann, The Geometry of Geodesics [Russian translation], Fizmatgiz, Moscow (1962). · Zbl 0112.37202
[5] K. Leichtweiss, Convex Sets [Russian translation], Nauka, Moscow (1985).
[6] V. N. Berestovskii, ”Homogeneous manifolds with intrinsic metric. II,” Sibirsk. Mat. Zh.,30, No. 2, 14–28 (1989).
[7] V. N. Berestovskii, Homogeneous Spaces with Intrinsic Metric [in Russian], Diss. Dokt. Fiz.-Mat. Nauk, Inst. Mat. (Novosibirsk), Novosibirsk (1990).
[8] A. D. Aleksandrov, ”über eine Verallgemeinerung der Riemannschen Geometrie,” Schrift. Inst. Math. der Deutschen Acad. Wiss., No. 1, 33–84 (1957).
[9] A. M. Vershik and V. Ya. Gershkovich, ”Nonholonomic dynamical systems. Geometry of distributions and variational problems,” in: Sovrem. Probl. Mat. Fund. Naprav. (Itogi Nauki i Tekhniki),16 [in Russian], VINITI, Moscow, 1987, pp. 5–85. · Zbl 0797.58007
[10] L. S. Kirillova, ”Non-Riemmannian metrics and the maximum principle,” Dokl. Akad. Nauk UzSSR, No. 7, 9–11 (1986).
[11] A. M. Vershik and O. A. Granichina, ”Reduction of nonholonomic variational problems to isoperimetric problems and connections in principal bundles,” Mat. Zametki,49, No. 5, 37–44 (1991). · Zbl 0734.49023
[12] R. Montgomery, Shortest Loops with a Fixed Holonomy [Preprint], MSRI (1988).
[13] V. N. Berestovskii, ”Submetries of space forms of nonnegative curvature,” Sibirsk. Mat. Zh.,28, No. 4, 44–56 (1987).
[14] Z. Ge, ”On a constrained variational problem and the spaces of horizontal paths,” Pacific J. Math.,149, No. 1, 61–94 (1991). · Zbl 0691.58021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.